Associative unital algebra

An associative unital algebra over a commutative unital ring is a monoid internal to the category of modules over that ring with respect to the usual tensor product of such modules.

Definition

Let $R$ be a commutative unital ring. All $R$ -modules considered are supposed to be unital. Let $\otimes$ be a tensor product functor for $R$ -modules and let $\alpha$ be the corresponding associator $(\cdot_1\otimes \cdot_2)\otimes \cdot_3\to \cdot_1\otimes(\cdot_2\otimes \cdot_3)$ and $\ell: R\otimes (\cdot)\to (\cdot)$ and $r:(\cdot)\to R\otimes (\cdot)$ the left and right unitors.

An associative unital algebra over $R$ is any triple $(A,m,\eta)$ such that

$A$ is an $R$ -module,
$m$ is an $R$ -module morphism $A\otimes A\to A$ (multiplication map),
$\eta$ is an $R$ -module morphism $R\to A$ (unit map)

and such that the following conditions are met:

$m\circ (\mathrm{id}_A\otimes m) \circ \alpha_{A,A,A}=m\circ (m\otimes \mathrm{id}_A)$ (associativity axiom),
$m\circ (\eta\otimes \mathrm{id}_A)=\ell_A$ (left unit axiom),
$m\circ (\mathrm{id}_A\otimes \eta)=r_A$ (right unit axiom).